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* Department of Animal and Dairy Science, University of Georgia, Athens 30602-2771;
and
University of Viçosa, Viçosa, MG 36571-000, Brazil; and
and
Embrapa Beef Cattle, Campo Grande-MS, CEP 79106-970, Brazil
2 Correspondence:
phone: 706-542-0951; E-mail:
ignacy{at}uga.edu.
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Abstract |
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 Top Abstract IntroductionMaterials and MethodsResults and DiscussionImplicationsLiterature Cited |
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Key Words: Beef Cattle • Genetic Parameters • Maximum Likelihood • Regression Analysis
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Introduction |
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TopAbstractIntroductionMaterials and MethodsResults and DiscussionImplicationsLiterature Cited |
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The application of longitudinal models for growth (Varona et al., 1997; Meyer, 1999; Villalba, 2000) allows use of all available records; therefore, preadjustment to constant ages is not needed. However, such models are complicated and hard to compute. In parameter estimation with random regression models, parameters corresponding to the extremes of trajectories or where the data are sparse may be poor (Meyer, 1999). If the parameters of the random regression models (RRM) are poor, the evaluation with this model may be less accurate than with multiple trait-models (MTM). One methodology for assessing the quality of parameters in RRM is to compare estimates obtained by RRM with estimates from MTM. Although the MTM estimates may be biased or less accurate compared to the underlying model due to preadjustments, these estimates are less likely to be affected by extremes of trajectories.
Albuquerque and Meyer (2001) looked at estimates of (co)variances using RRM in Nellore beef cattle. Computations were simplified by assuming that direct-maternal correlation was zero. Artifacts of RRM were visible especially for later ages, and parameters varied strongly among samples. Parameters by RRM were compared with univariate but not MTM analyses.
The purpose of this study was to obtain genetic parameters for sequential growth of beef cattle using RRM with data sets with different structures and to compare these estimates with those obtained by MTM.
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Materials and Methods |
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TopAbstractIntroductionMaterials and MethodsResults and DiscussionImplicationsLiterature Cited |
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Traits considered were birth weight (BW) and up to eight sequential weights at (approximate) ages of 60, 152, 243, 333, 426, 517, 601, and 683 d. Edits included eliminating records of animals outside the range of three standard deviations from the overall mean for each weight, and eliminating records ±50 d outside of ages listed above. Table 1
presents characteristics of the data.
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Two data sets were formed by sampling complete herds: one (MISS) from all herds without regard for the distribution of missing traits, and one (NMISS) from herds with no missing traits. To maximize connections, only larger herds were sampled. The MISS data set was obtained from sampling from herds with more than 500 BW records and an average contemporary group size for BW greater than five records within each herd. The NMISS data set was designed to evaluate the effect of missing traits on parameter estimates particularly in RRM; it was obtained from herds with complete records, more than 50 BW records, and an average contemporary group size for BW greater than five within each herd. Single-animal contemporary groups were eliminated from both samples, and then 5% of the herds that remained were sampled. Connectedness among herds was not verified because most cows were AI-sired, and therefore good connectedness was assumed. The number of animals in the pedigree file was 21,055 and 16,161 for MISS and NMISS, respectively. Both samples are characterized in Table 2.
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where yijklt = tth weight preadjusted for age in contemporary group i, age-of-dam class j of animal k and dam l; cgit = fixed effects of contemporary group i and weight t; aodjt = fixed effects of age of dam class j and weight t; dkt = random direct genetic effect of animal k and weight t; mlt and mpelt = random maternal genetic and maternal permanent environmental effect of dam l and weight t; and eijklt = random residual effect. The direct and maternal genetic effects were assumed correlated. This model can be written in matrix notation as:
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where y = vector of records preadjusted to fixed age; ß = vector of fixed effects (contemporary group and age of dam class); d = vector of additive direct genetic random effects of the animal; m = vector of additive maternal genetic random effects; mp = vector of random effects of maternal permanent environment; e = vector of residual random effects; X was the incidence matrix for fixed effects; and Z1, Z2, and Z3 were incidence matrices for animal, maternal, and maternal permanent environmental effects, respectively.
The variances and covariances were defined as follows:
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where G0d was a covariance matrix of random direct genetic effects; G0m was a covariance matrix of random maternal genetic effects; G0dm was a matrix of genetic covariances between direct and maternal effects; G0mp was a covariance matrix of random maternal permanent environmental effects; R0 was a covariance matrix of random residual effects; A was the additive genetic relationship matrix; Ic was an identity matrix whose order was the number of dams; In was an identity matrix whose order was the number of animals; and 
was the direct product operator.
(Co)variance components were estimated for five traits at a time. Parameters presented here were based on averages from analyses of models that contained that particular parameter.
The RRM was defined as follows:
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where yijklm = observation in contemporary group i, age of dam class j, animal k, dam l, and record m; caddj = fixed regression coefficient d for age of dam class j; ddk and pdk = random regression coefficients d for additive direct and permanent environmental effects of animal k; mdl and mpdl = random regression coefficients d for additive maternal and maternal permanent environmental effects of dam l; rdm = random regression coefficient d for error effects of record m; zdm = dth coefficient of Legendre polynomial for age at observation m; and 
ijklm = the residual effect. The purpose of the error effect was to indirectly model heterogeneous residual variance (Van der Werf and Schaeffer, 1997); the available software did not allow modeling this directly.
The random regression model could be written in matrix notation as:
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where yR was the vector of records; ßR was the vector of fixed effects; dR, p, mR, mpR, and r were vectors for additive direct genetic, permanent environmental, additive maternal genetic, maternal permanent environmental, and error effects, respectively; XR was the incidence matrix for fixed effects; ZRd, ZRm, and ZRr were incidence covariate matrices for direct, maternal, and error effects, respectively; and eR was a vector of residual effects.
The variances were defined as follows:
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where GR0d, GR0m, and GR0dm were 4 x 4 covariance matrices of random regression for direct effect, maternal genetic effect, and their covariances, respectively; G0p, GR0mp, and RR0 were 4 x 4 covariance matrices of random regression for permanent environmental, maternal permanent environmental, and error effects, respectively; 
was assumed constant residual variance; A was an additive genetic relationship matrix; Ik was an identity matrix whose order was the number of animals; Il was an identity matrix whose order was the number of dams; and In was an identity matrix whose order was the number of records.
For observation m containing trait t, and with Legendre polynomials corresponding to age for trait t, the residual effect in MTM is approximately equivalent to the sum of residual, permanent environmental, and error effects in RRM:
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Define z(t) as a set of polynomials corresponding to trait t. Then:
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and for combinations of traits t1 and t2:
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Multiple-Trait and Random Regression Models
The MTM is less structured than the RRM. It allows for any values of correlations among combinations of traits and effects. Because each trait is defined as an interval of ages, genetic correlations between traits in adjacent intervals may be less accurate as point estimates, but genetic correlations between traits corresponding to distant intervals may be quite accurate. Fluctuations in the form of sampling variance can be expected from trait to trait with the magnitude of fluctuation corresponding to the amount of data available to estimate a particular (co)variance. However, fluctuations of different parameters especially for distant traits are likely to be relatively independent. Therefore, trends with ages obtained by smoothing estimates by MTM may be satisfactory.
The RRM is structured by the use of estimating functions, which in this article were cubic Legendre polynomials. A covariance matrix in RRM allows for modeling of changes of variances along age and changes in genetic correlations among ages. However, the relationship between one parameter of RRM and a particular genetic (co)variance may be very complex because the same parameter may affect curves of several variances and correlations. Polynomials may result in insufficient ability to fit all the (co)variance curves well if the order is too low. Polynomials may result in modeling fluctuations as real curves if the order is too high. Uneven distribution of data may result in bad fit for some curves. Therefore, RRM are susceptible to artifacts, and their parameters need to be examined before any practical application. Cubic Legendre polynomials were selected in this study as a compromise between fitting capability, reduction of artifacts, and computational costs.
(Co)variance components for MTM and RRM were estimated by program REMLF90 (Misztal, 2001), which used an accelerated expectation maximization (EM) REML procedure.
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Results and Discussion |
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TopAbstractIntroductionMaterials and MethodsResults and DiscussionImplicationsLiterature Cited |
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, and the corresponding graphs are in Figure 1. For MTM, the estimates generally exhibit expected trends, with additive genetic and residual variances increasing with age and maternal variances increasing until the approximate age when calves are weaned and then decreasing for later ages. The shapes of the graphs were generally similar between the two data sets, with the exception of the last trait. An obvious explanation for differences between MISS and NMISS for variance estimates for the last trait is that only 3% of animals had records in MISS for that trait. Albuquerque and Meyer (2001), who analyzed a similar data set, discarded records after 630 d.
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The additive maternal variances were generally similar for both data sets. However, estimates of maternal permanent environment were larger for MISS except for BW, and estimates of direct-maternal covariances at ages 
152 d were larger negative values for NMISS. These differences could be due to different data structure in both data sets. According to R. L. Quaas (personal communication), estimates of maternal effects may be inaccurate if few sires are also maternal grandsires; most connections between the direct and maternal effects are provided by male relationships. This was the case for MISS but not for NMISS. Also, Meyer (1992a) has found a strong influence of data structure on the estimate of direct-maternal correlation.
Estimates of the residual variance were larger for MISS than for NMISS for all traits except BW. This also could be due to selection of data in MISS. However, differences for BW are difficult to explain except by assuming heterogeneous variances between MISS and NMISS.
The general pattern of the shape of the curves with RRM was similar to those with MTM from BW (1 d) through weight at 601 d of age. However, except for BW, the estimates of variances were generally larger, and the direct-maternal covariance estimates were more negative from RRM than from MTM. The most apparent differences in estimates from MTM were large increases in several of the estimates for RRM from MISS after about 600 d. Parameters estimated via RRM are susceptible to large sampling errors and estimation artifacts for data points along the trajectory that have small amounts of data (Van der Werf and Schaeffer, 1997). Estimates from RRM for NMISS did not exhibit such large increases. The BW estimates for additive direct and maternal variances and direct-maternal covariance were smaller when estimated using RRM than MTM. In particular, the BW additive direct variance estimate with RRM for MISS was only 36% as large as the MTM estimate in the same data set. Whereas estimates by MTM generally reflect averages of ages while they are points for RRM, estimates for BW have the same definition under both models.
Totals of permanent environmental, residual, and error effects in RRM approximate the estimates of the residual effect in MTM for NMISS, but not for MISS, where the estimates of the error effect seem to cycle and have multiple peaks at approximately 150, 450, and 680 d. Whereas the influence of missing traits seems to be most obvious for other effects at ages over 500 d, the error effect seems to be affected at earlier ages. This could be partially due to implementing heterogeneous variances for RRM via the error effect in this study; a direct implementation of heterogeneous residual variances may result in fewer artifacts.
Estimates of direct correlations at any age and at 1 (BW), 333, or 683 d obtained with MTM are shown in Figure 2. For MTM, the correlations between BW and later ages dropped below 0.6 at about 150 d of age for MISS and to about 0.4 for NMISS, indicating that BW is a different trait from later growth. Robinson (1996) reported direct genetic correlations between BW and later weights that varied from 0.52 to 0.59, and Meyer (1993) found direct genetic correlations between BW and later weights to range from 0.44 to 0.67.
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shows the additive direct genetic correlations for specific points. For MTM, the correlations between weight at 333 d and weight at 243 to 426 d were above 0.80 for both MISS and NMISS, which indicates that weights in the approximate range of 250 to 450 d could be adjusted to 333 d and analyzed as the same trait. Meyer (1993) found that estimates of direct genetic variance between weaning, yearling and final weights were close to 1.0. For MISS, the direct genetic correlations between 683 d and below 600 d of ages were below 0.60, suggesting that weight at 683 d cannot be well predicted from weight at earlier ages. The same correlation for NMISS was lower than 0.9, suggesting that later growth can be well predicted by earlier growth. Also, the pattern of these correlations with NMISS is much smoother. The completeness of the data strongly influences estimates of direct genetic correlations among ages. The pattern of correlations with RRM is very similar to those with MTM.
Estimates of maternal correlations are shown in Figure 3. For MTM, the estimates of additive maternal genetic correlations were within the range of previous reports (Waldron et al., 1993; Eler et al., 1995). However, these estimates were lower than those reported by Robinson (1996). Similar to estimates for direct correlations, the estimates for MISS followed a more irregular pattern than NMISS, particularly for correlations involving ages greater than 333 d. For NMISS, the maternal correlations between BW and days over 60 were lower than 0.20, suggesting that very little genetic relationship exists between maternal genetic control of fetal growth and milk production. Also for NMISS, the maternal correlations between weights from 243 to 601 d of ages were all higher than 0.70, indicating that maternal effects at later ages are primarily governed by the same genes. As in correlations for the direct effects, completeness of the data strongly influences estimates of direct genetic correlations among ages. The maternal correlations by RRM generally followed the same pattern as MTM; however estimates from RRM tended to be higher and smoother with MISS. Smoothing is partially due to lack of degrees of freedom in RRM to follow rapid changes.
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. All correlations were negative for all ages from both samples. These results were in agreement with the majority of reports in the literature (Bertrand and Benyshek, 1987; Garrick et al., 1989; Eler et al., 1995, De Mattos et al., 2000). For MTM with MISS, the correlations were more negative at early ages, then dropped to a low value and subsequently increased. For MTM with NMISS, the shape of correlations was very different and more negative. In NMISS, only about 10% of sires were maternal grandsires vs. approximately 50% in MISS. This must have caused higher sampling variance associated with the direct-maternal correlation. For RRM, the estimates were more negative than for MTM. Also, estimation artifacts may have been occurring. In particular, for MISS, the correlation at about 50 d reaches -1.0, which is unrealistic. One possibility for the seeming imprecision of the direct-maternal correlations with RRM is that, of all the correlations, the direct-maternal correlations have the smallest amount of information available. Therefore, these estimates may have a large sampling error. The value of direct-maternal correlation is of little importance after about 300 d because of small values of maternal heritability for later ages.
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. For MTM, the results were different since heritability estimates of BW for MISS were higher than those for NMISS, and the maternal heritability reached a peak at about 150 d for NMISS but not for MISS. Since the differences are more pronounced at earlier ages where the proportion of missing records is low or none, these differences may indicate heterogeneous herd variances or different management style between herds included in MISS and those included in NMISS. Also, it may be that additive direct and additive maternal heritability estimates for BW with MTM may be biased, particularly for NMISS, because producers submit a breed average estimate for BW when BW are not actually taken. For RRM, the heritability estimates were very different for BW and also for later ages with NMISS. Variances for growth may change as much as 200 times from BW to 683 d. A good fit for all the ages is attempted by RRM. It is possible that the fit is the worst for traits with the smallest variances. Differences for later ages with RRM and MISS are due to few records available for those ages.
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The results of analyses using the two data samples were quite different. Whereas parameters obtained from NMISS were generally less erratic than from MISS, they also resulted in values of direct-maternal genetic correlations that approached the unrealistic value of -1.0. The estimates from NMISS showed artifacts for older ages and large differences between genetic correlations among earlier and later ages. An improvement could be obtained by using a data set with few missing traits and good connections between the direct and maternal effects. Results in this study should be viewed as trends rather than absolute values, and no definite "true" parameters should be expected. When many samples of the same population are analyzed, sampling variance may be quite large (i.e., as found by De Mattos et al., 2000).
Growth curves in this study by RRM with MISS showed large increases of variance at later ages. It is likely that real data sets have many missing traits, and that data without missing traits are obtained by elimination of incomplete records. Use of selected data in analyses may result in selection bias. If an evaluation by RRM is desired, estimates of parameters by RRM may not be satisfactory.
One question in this study is whether cubic polynomials provided sufficient fit for RRM although the application of higher order polynomials is not possible because of excessive costs. The assumption of non-zero direct maternal correlation resulted in 8 x 8 (co)variance matrices for the genetic component, or as large as required by seventh order polynomials if zero correlation was assumed. After examining graphs and tables for RRM, it seems that the cubic regressions allowed for sufficiently realistic modeling of curves of variances and correlations for the direct and maternal effects, and that artifacts were mainly due to special data structure. One possible exception was direct-maternal correlation, which seemed to be inflated with RRM as compared to MTM for both data sets. That cubic fit was sufficient is partially supported by presence of at least one very small eigenvalue (with value <0.01% of the largest eigenvalue) in each random effect.
Several strategies may be used for obtaining "better" estimates. One could be to use larger, more carefully selected data. Another strategy would be to use functions other than polynomials that are less susceptible to artifacts (i.e., fractional polynomials as applied by Robert-Granie et al., 2002). In yet another strategy, multiple-trait (MT) parameters could be "smoothed" and converted to an RRM scale (Lidauer and Mäntysaari, 1999). Finally, the parameters by RRM could be constructed based on estimates from MT and RRM models, and literature information (i.e., as in Misztal et al., 2000). With MT methodology, parameters estimated for BW are optimal because that trait is obtained at the same age and subsequently no preadjustment is necessary. To ensure that evaluation of BW by RRM is optimal, it may be necessary to "adjust" RRM parameters so they would equal those of MT for BW.
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Implications |
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TopAbstractIntroductionMaterials and MethodsResults and DiscussionImplicationsLiterature Cited |
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Footnotes |
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Received for publication April 22, 2002. Accepted for publication December 24, 2002.
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Literature Cited |
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TopAbstractIntroductionMaterials and MethodsResults and DiscussionImplicationsLiterature Cited |
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) and no missing (
) traits and for the random regression models with missing (—) and no missing (—) traits.